Minimal-entropy martingale measure

Short description: Method used to minimise uncertainty between probabilities

In probability theory, the minimal-entropy martingale measure (MEMM) is the risk-neutral probability measure that minimises the entropy difference between the objective probability measure, [math]\displaystyle{ P }[/math], and the risk-neutral measure, [math]\displaystyle{ Q }[/math]. In incomplete markets, this is one way of choosing a risk-neutral measure (from the infinite number available) so as to still maintain the no-arbitrage conditions.

The MEMM has the advantage that the measure [math]\displaystyle{ Q }[/math] will always be equivalent to the measure [math]\displaystyle{ P }[/math] by construction. Another common choice of equivalent martingale measure is the minimal martingale measure, which minimises the variance of the equivalent martingale. For certain situations, the resultant measure [math]\displaystyle{ Q }[/math] will not be equivalent to [math]\displaystyle{ P }[/math].

In a finite probability model, for objective probabilities [math]\displaystyle{ p_i }[/math] and risk-neutral probabilities [math]\displaystyle{ q_i }[/math] then one must minimise the Kullback–Leibler divergence [math]\displaystyle{ D_{KL}(Q\|P) = \sum_{i=1}^N q_i \ln\left(\frac{q_i}{p_i}\right) }[/math] subject to the requirement that the expected return is [math]\displaystyle{ r }[/math], where [math]\displaystyle{ r }[/math] is the risk-free rate.

References

• M. Frittelli, Minimal Entropy Criterion for Pricing in One Period Incomplete Markets, Working Paper. University of Brescia, Italy (1995).